Question

Find the point, M, that divides segment AB into a ratio of 4:7 if A is at (-33, 0) and B is at (0, 44).

Answer 1

Let's say the point is C, and divides AB on a 4:7 ratio, from A to B, meaning the AC segment takes 4 units of the lot and the CB segment takes 7.

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(-33,0)\qquad B(0,44)\qquad \qquad \stackrel{\textit{ratio from A to B}}{4:7} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{4}{7}\implies \cfrac{A}{B} = \cfrac{4}{7}\implies 7A=4B\implies 7(-33,0)=4(0,44)\\\\ -------------------------------$

$\bf C=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\ -------------------------------\\\\ C=\left(\cfrac{(7\cdot -33)+(4\cdot 0)}{4+7}\quad ,\quad \cfrac{(7\cdot 0)+(4\cdot 44)}{4+7}\right) \\\\\\ C=\left( \cfrac{-231+0}{11}~~,~~\cfrac{0+176}{11} \right)\implies C=(-21~,~16)$